b: \(\left(x-\dfrac{1}{3}\right)^3=-\dfrac{8}{27}\)

=>\(\left(x-\dfrac{1}{3}\right)^3=\left(-\dfrac{2}{3}\right)^3\)

=>\(x-\dfrac{1}{3}=-\dfrac{2}{3}\)

=>\(x=-\dfrac{2}{3}+\dfrac{1}{3}=-\dfrac{1}{3}\)

c: \(\left(5x+1\right)^2=\dfrac{36}{49}\)

=>\(\left[{}\begin{matrix}5x+1=\dfrac{6}{7}\\5x+1=-\dfrac{6}{7}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{6}{7}-1=-\dfrac{1}{7}\\5x=-\dfrac{6}{7}-1=-\dfrac{13}{7}\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x=-\dfrac{1}{7}:5=-\dfrac{1}{35}\\x=-\dfrac{13}{7}:5=-\dfrac{13}{35}\end{matrix}\right.\)

d: \(\left(\dfrac{1}{3}-\dfrac{3}{2}x\right)^2=2\dfrac{1}{4}\)

=>\(\left(\dfrac{3}{2}x-\dfrac{1}{3}\right)^2=\dfrac{9}{4}\)

=>\(\left[{}\begin{matrix}\dfrac{3}{2}x-\dfrac{1}{3}=\dfrac{3}{2}\\\dfrac{3}{2}x-\dfrac{1}{3}=-\dfrac{3}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\dfrac{3}{2}x=\dfrac{3}{2}+\dfrac{1}{3}=\dfrac{11}{6}\\\dfrac{3}{2}x=-\dfrac{3}{2}+\dfrac{1}{3}=-\dfrac{7}{6}\end{matrix}\right.\)

=>\(\left[{}\begin{matrix}x=\dfrac{11}{6}:\dfrac{3}{2}=\dfrac{11}{6}\cdot\dfrac{2}{3}=\dfrac{11}{9}\\x=-\dfrac{7}{6}:\dfrac{3}{2}=-\dfrac{7}{6}\cdot\dfrac{2}{3}=-\dfrac{7}{9}\end{matrix}\right.\)

e: \(\left(\dfrac{4}{5}\right)^{2x+5}=\dfrac{256}{625}\)

=>\(\left(\dfrac{4}{5}\right)^{2x+5}=\left(\dfrac{4}{5}\right)^4\)

=>2x+5=4

=>2x=4-5=-1

=>\(x=-\dfrac{1}{2}\)

g: \(\left(\dfrac{1}{3}\right)^{x+1}+\left(\dfrac{1}{3}\right)^{x+2}=\dfrac{1}{12}\)

=>\(\left(\dfrac{1}{3}\right)^x\cdot\dfrac{1}{3}+\left(\dfrac{1}{3}\right)^x\cdot\dfrac{1}{9}=\dfrac{1}{12}\)

=>\(\left(\dfrac{1}{3}\right)^x\left(\dfrac{1}{3}+\dfrac{1}{9}\right)=\dfrac{1}{12}\)
=>\(\left(\dfrac{1}{3}\right)^x=\dfrac{1}{12}:\dfrac{4}{9}=\dfrac{1}{12}\cdot\dfrac{9}{4}=\dfrac{3}{4\cdot4}=\dfrac{3}{16}\)

=>\(x=log_{\dfrac{1}{3}}\left(\dfrac{3}{16}\right)\)

Mọi người giúp mk bài này với!

          Đây là toán nâng cao chuyên đề bội ước, cấu trúc thi chuyên, thi học sinh giỏi các cấp. Hôm nay, Olm sẽ hướng dẫn các em giải chi tiết dạng này như sau. 

          45 = 3².5¹

Vì 45 là ước  chung lớn nhất của A = 3^a.5³ và 3³.5^b

Nên \(\left\{{}\begin{matrix}3^a=3^2\\5^b=5^1\end{matrix}\right.\)

        ⇒ \(\left\{{}\begin{matrix}a=2\\b=1\end{matrix}\right.\)

Vậy a + b = 2 + 1 = 3

a: \(\left(\sqrt{\dfrac{4}{3}}+\sqrt{3}\right)\cdot\sqrt{6}\)

\(=\sqrt{\dfrac{4}{3}\cdot6}+\sqrt{3\cdot6}\)

\(=\sqrt{8}+\sqrt{18}=2\sqrt{2}+3\sqrt{2}=5\sqrt{2}\)

b: \(\left(1-2\sqrt{5}\right)^2=\left(2\sqrt{5}-1\right)^2\)

\(=\left(2\sqrt{5}\right)^2-2\cdot2\sqrt{5}\cdot1+1\)

\(=21-4\sqrt{5}\)

c: \(2\sqrt{3}-\sqrt{27}=2\sqrt{3}-3\sqrt{3}=-\sqrt{3}\)

d: \(\sqrt{45}-\sqrt{20}+\sqrt{5}\)

\(=3\sqrt{5}-2\sqrt{5}+\sqrt{5}\)

\(=4\sqrt{5}-2\sqrt{5}=2\sqrt{5}\)

\(P=\left(1+\dfrac{1}{\sqrt{x}}\right)\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}-1}-\dfrac{2}{x-1}\right)\)

\(=\dfrac{\sqrt{x}+1}{\sqrt{x}}\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{1}{\sqrt{x}-1}-\dfrac{2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\dfrac{\sqrt{x}+1}{\sqrt{x}}\cdot\dfrac{\sqrt{x}-1+\sqrt{x}+1-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{2\sqrt{x}-2}{\left(\sqrt{x}-1\right)\cdot\sqrt{x}}=\dfrac{2\left(\sqrt[]{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}=\dfrac{2}{\sqrt{x}}\)

1: Thay x=9 vào A, ta được:

\(A=\dfrac{3\cdot3}{3+2}=\dfrac{9}{5}\)

2: \(B=\dfrac{x+4}{x-4}-\dfrac{2}{\sqrt{x}-2}\)

\(=\dfrac{x+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}-\dfrac{2}{\sqrt{x}-2}\)

\(=\dfrac{x+4-2\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{x-2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{\sqrt{x}}{\sqrt{x}+2}\)

3: \(A-B< \dfrac{3}{2}\)

=>\(\dfrac{3\sqrt{x}}{\sqrt{x}+2}-\dfrac{\sqrt{x}}{\sqrt{x}+2}< \dfrac{3}{2}\)

=>\(\dfrac{2\sqrt{x}}{\sqrt{x}+2}-\dfrac{3}{2}< 0\)

=>\(\dfrac{4\sqrt{x}-3\left(\sqrt{x}+2\right)}{2\left(\sqrt{x}+2\right)}< 0\)

=>\(\dfrac{\sqrt[]{x}-6}{2\left(\sqrt{x}+2\right)}< 0\)

=>\(\sqrt{x}-6< 0\)

=>\(\sqrt{x}< 6\)

=>0<=x<36

mà x là số nguyên dương lớn nhất thỏa mãn

nên x=35

a: \(P=\left(\dfrac{1}{\sqrt{a}-1}-\dfrac{1}{\sqrt{a}}\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-2}-\dfrac{\sqrt{a}+2}{\sqrt{a}-1}\right)\)

\(=\dfrac{\sqrt{a}-\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}:\dfrac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)-\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}\)

\(=\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\cdot\dfrac{\left(\sqrt{a}-2\right)\left(\sqrt{a}-1\right)}{a-1-\left(a-4\right)}\)

\(=\dfrac{\sqrt{a}-2}{3\sqrt{a}}\)

b: P>1/6

=>P-1/6>0

=>\(\dfrac{\sqrt{a}-2}{3\sqrt{a}}-\dfrac{1}{6}>0\)

=>\(\dfrac{6\left(\sqrt{a}-2\right)-3\sqrt{a}}{18\sqrt{a}}>0\)

=>\(6\left(\sqrt{a}-2\right)-3\sqrt{a}>0\)

=>\(3\sqrt{a}-12>0\)

=>\(\sqrt{a}>4\)
=>a>16

16¹⁹ và 8²⁵

16¹⁹ = (2⁴)¹⁹ = 2⁷⁶

8²⁵ = (2³)²⁵ = 2⁷⁵

Vì 2⁷⁵ < 2⁷⁶ nên 

16¹⁹ > 8²⁵

a: \(A=\left(\dfrac{4\sqrt{x}}{\sqrt{x}+2}+\dfrac{8x}{4-x}\right):\left(\dfrac{\sqrt{x}-1}{x-2\sqrt{x}}-\dfrac{2}{\sqrt{x}}\right)\)

\(=\left(\dfrac{4\sqrt{x}}{\sqrt{x}+2}-\dfrac{8x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right):\left(\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-2\right)}-\dfrac{2}{\sqrt{x}}\right)\)

\(=\dfrac{4\sqrt{x}\left(\sqrt{x}-2\right)-8x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}:\dfrac{\sqrt{x}-1-2\left(\sqrt{x}-2\right)}{\sqrt[]{x}\left(\sqrt{x}-2\right)}\)

\(=\dfrac{4x-8\sqrt{x}-8x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\cdot\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)}{\sqrt{x}-1-2\sqrt{x}+4}\)

\(=\dfrac{-4x-8\sqrt{x}}{\sqrt{x}+2}\cdot\dfrac{\sqrt{x}}{-\sqrt{x}+3}=\dfrac{-4\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)}\cdot\dfrac{-\sqrt{x}}{\sqrt{x}-3}\)

\(=\dfrac{4x}{\sqrt{x}-3}\)

b: A=-2

=>\(4x=-2\left(\sqrt{x}-3\right)=-2\sqrt{x}+6\)

=>\(4x+2\sqrt{x}-6=0\)

=>\(2x+\sqrt{x}-3=0\)

=>\(\left(2\sqrt{x}+3\right)\left(\sqrt{x}-1\right)=0\)

mà \(2\sqrt{x}+3>=3>0\forall x\) thỏa mãn ĐKXĐ

nên \(\sqrt{x}-1=0\)

=>x=1(nhận)

a: \(Q=\dfrac{\sqrt{x^3}-\sqrt{x}+2x-2}{\sqrt{x}+2}\)

\(=\dfrac{x\sqrt{x}-\sqrt{x}+2\left(x-1\right)}{\left(\sqrt{x}+2\right)}\)

\(=\dfrac{\sqrt{x}\left(x-1\right)+2\left(x-1\right)}{\sqrt{x}+2}=\dfrac{\left(x-1\right)\left(\sqrt{x}+2\right)}{\sqrt{x}+2}=x-1\)

\(P=\dfrac{2x-3\sqrt{x}-2}{\sqrt{x}-2}\)

\(=\dfrac{2x-4\sqrt{x}+\sqrt{x}-2}{\sqrt{x}-2}\)

\(=\dfrac{2\sqrt{x}\left(\sqrt{x}-2\right)+\left(\sqrt{x}-2\right)}{\sqrt{x}-2}=2\sqrt{x}+1\)

b: P=Q

=>\(x-1=2\sqrt{x}+1\)

=>\(x-2\sqrt{x}-2=0\)

=>\(x-2\sqrt{x}+1=3\)

=>\(\left(\sqrt{x}-1\right)^2=3\)

mà \(\sqrt{x}-1>=-1\) với mọi x thỏa mãn ĐKXĐ

nên \(\sqrt{x}-1=\sqrt{3}\)

=>\(\sqrt{x}=1+\sqrt{3}\)

=>\(x=\left(1+\sqrt{3}\right)^2=4+2\sqrt{3}\left(nhận\right)\)

a: \(P=\dfrac{3}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-5}{x-1}\)

\(=\dfrac{3}{\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}-\dfrac{\sqrt{x}-5}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{3\left(\sqrt{x}-1\right)-\sqrt{x}-1-\sqrt{x}+5}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)

\(=\dfrac{3\sqrt{x}-3-2\sqrt{x}+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{1}{\sqrt{x}-1}\)

b: \(x=24-16\sqrt{2}=8\left(3-2\sqrt{2}\right)=8\left(\sqrt{2}-1\right)^2\)

Thay \(x=8\left(\sqrt{2}-1\right)^2\) vào P, ta được:

\(P=\dfrac{1}{\sqrt{8\left(\sqrt{2}-1\right)^2}-1}\)

\(=\dfrac{1}{2\sqrt{2}\left(\sqrt{2}-1\right)-1}=\dfrac{1}{4-2\sqrt{2}-1}\)

\(=\dfrac{1}{3-2\sqrt{2}}=3+2\sqrt{2}\)